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Properties

Label 2-30-5.4-c5-0-5
Degree $2$
Conductor $30$
Sign $-0.983 - 0.178i$
Analytic cond. $4.81151$
Root an. cond. $2.19351$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s − 9i·3-s − 16·4-s + (−55 − 10i)5-s − 36·6-s + 4i·7-s + 64i·8-s − 81·9-s + (−40 + 220i)10-s − 500·11-s + 144i·12-s − 288i·13-s + 16·14-s + (−90 + 495i)15-s + 256·16-s − 1.51e3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.983 − 0.178i)5-s − 0.408·6-s + 0.0308i·7-s + 0.353i·8-s − 0.333·9-s + (−0.126 + 0.695i)10-s − 1.24·11-s + 0.288i·12-s − 0.472i·13-s + 0.0218·14-s + (−0.103 + 0.568i)15-s + 0.250·16-s − 1.27i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $-0.983 - 0.178i$
Analytic conductor: \(4.81151\)
Root analytic conductor: \(2.19351\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :5/2),\ -0.983 - 0.178i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0610391 + 0.676934i\)
\(L(\frac12)\) \(\approx\) \(0.0610391 + 0.676934i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
3 \( 1 + 9iT \)
5 \( 1 + (55 + 10i)T \)
good7 \( 1 - 4iT - 1.68e4T^{2} \)
11 \( 1 + 500T + 1.61e5T^{2} \)
13 \( 1 + 288iT - 3.71e5T^{2} \)
17 \( 1 + 1.51e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.34e3T + 2.47e6T^{2} \)
23 \( 1 + 4.10e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.64e3T + 2.05e7T^{2} \)
31 \( 1 + 5.61e3T + 2.86e7T^{2} \)
37 \( 1 - 7.28e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.89e4T + 1.15e8T^{2} \)
43 \( 1 + 2.40e3iT - 1.47e8T^{2} \)
47 \( 1 + 8.90e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.98e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.83e4T + 7.14e8T^{2} \)
61 \( 1 - 1.82e4T + 8.44e8T^{2} \)
67 \( 1 + 6.59e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.88e4T + 1.80e9T^{2} \)
73 \( 1 + 3.08e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.02e4T + 3.07e9T^{2} \)
83 \( 1 + 2.46e3iT - 3.93e9T^{2} \)
89 \( 1 + 2.26e4T + 5.58e9T^{2} \)
97 \( 1 - 3.69e4iT - 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.33684691231028347665896506160, −13.78417004913542272665459685489, −12.62919959806152997394507802360, −11.69765441709609269383390429165, −10.41228894171817791107109823065, −8.579693907971565890585010067035, −7.38081507658713994227223571280, −5.01471920066037633071233408324, −2.92010845776706537955686487508, −0.42637730941043051683689221875, 3.70449912625009833278375496598, 5.35821594259550053133719018635, 7.32508193059250942343220121050, 8.504346053018820821900824369230, 10.14998589676599840255360220479, 11.50688857556127183595983291963, 13.08611873048022032334843918437, 14.56949364546473535323718923453, 15.58424982272352289760708492770, 16.21640752056387997776416814780

Graph of the $Z$-function along the critical line